Zero knowledge proofs

The notion of proof is central to so many fields. In mathematics, we
want to prove that a certain assertion is correct. In other sciences, we
often want to accumulate a preponderance of evidence (or statistical
significance) to reject certain hypothesis. In criminal law the
prosecution famously needs to prove its case “beyond a reasonable
doubt”. Cryptography turns out to give some new twists on this ancient
notion.

Typically a proof that some assertion X is true, also reveals some
information about why X is true. When Hercule Poirot proves that
Norman Gale killed Madame Giselle he does so by showing how Gale
committed the murder by dressing up as a flight attendant and stabbing
Madame Gisselle with a poisoned dart. Could Hercule convince us beyond a
reasonable doubt that Gale did the crime without giving any information
on how the crime was committed? Can the Russians prove to the U.S.
that a sealed box contains an authentic nuclear warhead without
revealing anything about its design? Can I prove to you that the number
\(m=385,608,108,395,369,363,400,501,273,594,475,104,405,448,848,047,062,278,473,983\)
has a prime factor whose last digit is \(7\) without giving you any
information about \(m\)’s prime factors? We won’t answer the first
question, but will show some insights on the latter two.In case you are curious, the factors of \(m\) are
\(1,172,192,558,529,627,184,841,954,822,099\) and
\(328,963,108,995,562,790,517,498,071,717\).

Zero knowledge proofs are proofs that fully convince that a statement
is true without yielding any additional knowledge. So, after seeing a
zero knowledge proof that \(m\) has a factor ending with \(7\), you’ll be no
closer to knowing \(m\)’s factorization than you were before. Zero
knowledge proofs were invented by Goldwasser, Micali and Rackoff in 1982
and have since been used in great many settings. How would you achieve
such a thing, or even define it? And why on earth would it be useful?
This is the topic of this lecture.

Applications for zero knowledge proofs.

Before we talk about how to achieve zero knowledge, let us discuss some
of its potential applications:

Nuclear disarmament

The United States and Russia have reached a dangerous and expensive
equilibrium by which each has about 7000 nuclear
warheads,
much more than is needed to decimate each others’ population (and the
population of much of the rest of the world).To be fair, “only” about 170 million Americans live in the 50
largest metropolitan
areas
and so arguably many people will survive at least the initial impact
of a nuclear war, though it had been estimated that even a “small”
nuclear war involving detonation of 100 not too large warheads could
have devastating global
consequences. Having so many weapons
increases the chance of “leakage” of weapons, or of an accidental launch
(which can result in an all out war) through fault in communications or
rogue commanders. This also threatens the delicate balance of the
Non-Proliferation
Treaty
which at its core is a bargain where non-weapons states agree not to
pursue nuclear weapons and the five nuclear weapon states agree to make
progress on nuclear disarmament. These huge quantities of nuclear
weapons are not only dangerous, as they increase the chance of a leak or
of an individual failure or rogue commander causing a world catastrophe,
but also extremely expensive to maintain.

For all of these reasons, in 2009, U.S. President Obama called to set as
a long term goal a “world without nuclear weapons” and in 2012 talked
about concretely talking to Russia about reducing “not only
our strategic nuclear warheads, but also tactical weapons and warheads
in reserve”. On the other side, Russian President Putin has said already
in 2000 that he sees “no obstacles that could hamper future deep cuts of
strategic offensive armaments”. (Though as of 2018, political winds on
both sides have shifted away from disarmament and more toward armament.)

There are many reasons why progress on nuclear disarmament has been so
slow, and most of them have nothing to do with zero knowledge or any
other piece of technology. But there are some technical hurdles as well.
One of those hurdles is that for the U.S. and Russia to go beyond
restricting the number of deployed weapons to significantly reducing
the stockpiles, they need to find a way for one country to verifiably
prove that it has dismantled warheads. As mentioned in my work with
Glaser and
Goldston
(see also this
page), a key
stumbling block is that the design of a nuclear warhard is of course
highly classified and about the last thing in the world that the U.S.
would like to share with Russia and vice versa. So, how can the U.S.
convince the Russian that it has destroyed a warhead, when it cannot let
Russian experts anywhere near it?

Voting

Electronic voting has been of great interest for many reasons. One
potential advantage is that it could allow completely transparent vote
counting, where every citizen could verify that the votes were counted
correctly. For example, Chaum suggested an approach to do so by
publishing an encryption of every vote and then having the central
authority prove that the final outcome corresponds to the counts of
all the plaintexts. But of course to maintain voter privacy, we need to
prove this without actually revealing those plaintexts. Can we do so?

More applications

I chose these two examples above precisely because they are hardly the
first that come to mind when thinking about zero knowledge. Zero
knowledge has been used for many cryptographic applications. One such
application (originating from work of Fiat and Shamir) is the use for
identification protocols. Here Alice knows a solution \(x\) to a puzzle
\(P\), and proves her identity to Bob by, for example, providing an
encryption \(c\) of \(x\) and proving in zero knowledge that \(c\) is indeed
an encryption of a solution for \(P\).As we’ll see, technically what Alice needs to do in such a
scenario is use a zero knowledge proof of knowledge of a solution
for \(P\). Bob can verify the proof, but
because it is zero knowledge, learns nothing about the solution of the
puzzle and will not be able to impersonate Alice. An alternative
approach to such identification protocols is through using digital
signatures; this connection goes both ways and zero knowledge proofs
have been used by Schnorr and others as a basis for signature schemes.

Another very generic application is for “compiling protocols”. As we’ve
seen time and again, it is often much easier to handle passive
adversaries than active ones. (For example, it’s much easier to get
CPA security against the eavesdropping Eve than CCA security against the
person-in-the-middle Mallory.) Thus it would be wonderful if we could
“compile” a protocol that is secure with respect to passive attacks into
one that is secure with respect to active ones. As was first shown by
Goldreich, Micali, and Wigderson, zero knowledge proofs yield a very
general such compiler. The idea is that all parties prove in zero
knowledge that they follow the protocol’s specifications. Normally, such
proofs might require the parties to reveal their secret inputs, hence
violating security, but zero knowledge precisely guarantees that we can
verify correct behaviour without access to these inputs.

Defining and constructing zero knowledge proofs

So, zero knowledge proofs are wonderful objects, but how do we get them?
In fact, we haven’t answered the even more basic question of how do we
define zero knowledge? We have to start by the most basic task of
defining what we mean by a proof.

A proof system can be thought of as an algorithm \(V\) (for “verifier”)
that takes as input a statement which is some string \(x\) and another
string \(\pi\) known as the proof and outputs \(1\) if and only if \(\pi\)
is a valid proof that the statement \(x\) is correct. For example:

In Euclidean geometry, statements are geometric facts such as
“in any triangle the degrees sum to 180 degrees” and the proofs
are step by step derivations of the statements from the five basic
postulates.

In Zermelo-Fraenkel + Axiom of Choice
(ZFC)
a statement is some purported fact about sets (e.g., the Riemann
HypothesisIntegers can be coded as sets in various ways. For example, one
can encode \(0\) as \(\emptyset\) and if \(N\) is the set encoding \(n\), we
can encode \(n+1\) using the \(n+1\)-element set \(\{ N \} \cup N\).), and a proof is a step by step derivation of it
from the axioms.

We can many define other “theories”. For example, a theory where the
statements are pairs \((x,m)\) such that \(x\) is a quadratic residue
modulo \(m\) and a proof for \(x\) is the number \(s\) such that
\(x=s^2 \pmod{m}\), or a theory where the theorems are Hamiltonian
graphs \(G\) (graphs on \(n\) vertices that contain an \(n\)-long cycle)
and the proofs are the description of the cycle.

All these proof systems have the property that the verifying algorithm
\(V\) is efficient. Indeed, that’s the whole point of a proof \(\pi\)-
it’s a sequence of symbols that makes it easy to verify that the
statement is true.

To achieve the notion of zero knowledge proofs, Goldwasser and Micali
had to consider a generalization of proofs from static sequences of
symbols to interactive probabilistic protocols between a prover and a
verifier. Let’s start with an informal example. The vast majority of
humans have three types of cone cells in their eyes. This is the reason
why we perceive the sky as
blue
(see also
this),
despite its color being quite a different spectrum than the blue of the
rainbow, is that the projection of the sky’s color to our cones is
closest to the projection of blue. It has been suggested that a tiny
fraction of the human population might have four functioning cones (in
fact, only women, as it would require two X chromosomes and a certain
mutation). How would a person prove to another that she is a in fact
such a tetrachromat ?

Proof of tetrachromacy:

Suppose that Alice is a tetrachromat and can distinguish between the
colors of two pieces of plastic that would be identical to a
trichromat. She wants to prove to a trichromat Bob that the two pieces
are not identical. She can do this as follows:

Alice and Bob will repeat the following experiment \(n\) times: Alice
turns her back and Bob tosses a coin and with probability 1/2 leaves
the pieces as they are, and with probability 1/2 switches the right
piece with the left piece. Alice needs to guess whether Bob switched
the pieces or not.

If Alice is successful in all of the \(n\) repetitions then Bob will
have \(1-2^{-n}\) confidence that the pieces are truly different.

We now consider a more “mathematical” example along similar lines.
Recall that if \(x\) and \(m\) are numbers then we say that \(x\) is a
quadratic residue modulo \(m\) if there is some \(s\) such that
\(x=s^2 \pmod{m}\). Let us define the function \(NQR(m,x)\) to output \(1\) if
and only if \(x \neq s^2 \pmod{m}\) for every \(s \in \{0,\ldots, m-1\}\).
There is a very simple way to prove statements of the form
“\(NQR(m,x)=0\)”: just give out \(s\). However, here is an interactive proof
system to prove statements of the form “\(NQR(m,x)=1\)”:

We have two parties: Alice and Bob. The common input is
\((m,x)\) and Alice wants to convince Bob that \(NQR(m,x)=1\). (That is,
that \(x\) is not a quadratic residue modulo \(m\)).

We assume that Alice can compute \(NQR(m,w)\) for every
\(w\in \{0,\ldots,m-1\}\) but Bob is polynomial time.

The protocol will work as follows:

Bob will pick some random \(s\in \Z^*_m\) (e.g., by picking a random
number in \(\{1,\ldots,m-1\}\) and discard it if it has nontrivial
g.c.d. with \(m\)) and toss a coin \(b\in\{0,1\}\). If \(b=0\) then Bob
will send \(s^2 \pmod{m}\) to Alice and otherwise he will send
\(xs^2 \pmod{m}\) to Alice.

Alice will use her ability to compute \(NQR(m,\cdot)\) to respond with
\(b'=0\) if Bob sent a quadratic residue and with \(b'=1\) otherwise.

Bob accepts the proof if \(b=b'\).

To see that Bob will indeed accept the proof, note that if \(x\) is a
non-residue then \(xs^2\) will have to be a non-residue as well (since if
it had a root \(s'\) then \(s's^{-1}\) would be a root of \(x\)). Hence it
will always be the case that \(b'=b\).

Moreover, if \(x\) was a quadratic residue of the form \(x=s'^2 \pmod{m}\)
for some \(s'\), then \(xs^2=(s's)^2\) is simply a random quadratic residue,
which means that in this case Bob’s message is distributed the same
regardless of whether \(b=0\) or \(b=1\), and no matter what she does, Alice
has probability at most \(1/2\) of guessing \(b\). Hence if Alice is always
successful than after \(n\) repetitions Bob would have \(1-2^{-n}\)
confidence that \(x\) is indeed a non-residue modulo \(m\).

Please stop and make sure you see the similarities between this protocol
and the one for demonstrating that the two pieces of plastic do not have
identical colors.

Let us now make the formal definition:

Let \(f:\{0,1\}^* \rightarrow \{0,1\}\) be some function. A probabilistic
proof for \(f\) (i.e., a proof for statements of the form “\(f(x)=1\)”) is
a pair of interactive algorithms \((P,V)\) such that \(V\) runs in
polynomial time and they satisfy:

Completeness: If \(f(x)=1\) then on input \(x\), if \(P\) and \(V\) are
given input \(x\) and interact, then at the end of the interaction \(V\)
will output Accept with probability at least \(0.9\).

Soundness: If If \(f(x)=0\) then for any arbitrary (efficient or
non efficient) algorithm \(P^*\), if \(P^*\) and \(V\) are given input \(x\)
and interact then at the end \(V\) will output Accept with
probability at most \(0.1\)

In many texts proof systems are defined with respect to languages as
opposed to functions. That is, instead of talking about a function
\(f:\{0,1\}^* \rightarrow \{0,1\}\) we talk about a lanugage
\(L \subseteq \{0,1\}^*\). These two viewpoints are completely equivalent
via the mapping \(f \longleftrightarrow L\) where
\(L = \{ x \;| f(x) = 1 \}\).

Note that we don’t necessarily require the prover to be efficient (and
indeed, in some cases it might not be). On the other hand, our soundness
condition holds even if the prover uses a non efficient strategy.People have considered the notion of zero knowledge systems where
soundness holds only with respect to efficient provers; these are
known as argument systems. We
say that a proof system has an efficient prover if there is an NP-type
proof system \(\Pi\) for \(L\) (that is some efficient algorithm \(\Pi\) such
that there exists \(\pi\) with \(\Pi(x,\pi)=1\) iff \(x\in L\) and such that
\(\Pi(x,\pi)=1\) implies that \(|\pi|\leq poly(|x|)\)), such that the
strategy for \(P\) can be implemented efficiently given any static proof
\(\pi\) for \(x\) in this system.

Up until now, we always considered cryptographic protocols where Alice
and Bob trusted one another, but were worried about some adversary
controlling the channel between them. Now we are in a somewhat more
“suspicious” setting where the parties do not fully trust one another.
In such protocols there is always a “prescribed” or honest strategy
that a particular party should follow, but we generally don’t want the
other parties’ security to rely on someone else’s good intention, and
hence analyze also the case where a party uses an arbitrary
malicious strategy. We sometimes also consider the honest but
curious case where the adversary is passive and only collects
information, but does not deviate from the prescribed strategy.

Protocols typically only guarantee security for party A when it behaves
honestly - a party can always chose to violate its own security and
there is not much we can (or should?) do about it.

Defining zero knowledge

So far we merely defined the notion of an interactive proof system, but
we need to define what it means for a proof to be zero knowledge.
Before we attempt a definition, let us consider an example. Going back
to the notion of quadratic residuosity, suppose that \(x\) and \(m\) are
public and Alice knows \(s\) such that \(x=s^2 \pmod{m}\). She wants to
convince Bob that this is the case. However she prefers not to reveal
\(s\). Can she convince Bob that such an \(s\) exist without revealing any
information about it? Here is a way to do so:

Protocol ZK-QR: Public input for Alice and Bob: \(x,m\); Alice’s
private input is \(s\) such that \(x=s^2 \pmod{m}\).

If \(b=0\) then Alice reveals \(ss'\), hence giving out a root for \(x'\);
if \(b=1\) then Alice reveals \(s'\), hence showing a root for
\(x'x^{-1}\).

Bob checks that the value \(s''\) revealed by Alice is indeed a root
of \(x'x^{-b}\), if so then it “accepts” the proof.

If \(x\) was not a quadratic residue then no matter how \(x'\) was chosen,
either \(x'\) or \(x'x^{-1}\) is not a residue and hence Bob will reject
the proof with probability at least \(1/2\). By repeating this \(n\) times,
we can reduce the probability of Bob accepting a the proof of a non
residue to \(2^{-n}\).

On the other hand, we claim that we didn’t really reveal anything about
\(s\). Indeed, if Bob chooses \(b=0\), then the two messages \((x',ss')\) he
sees can be thought of as a random quadratic residue \(x'\) and its root.
If Bob chooses \(b=1\) then after dividing by \(x\) (which he could have
done by himself) he still gets a random residue \(x''\) and its root \(s'\).
In both cases, the distribution of these two messages is completely
independent of \(s\), and hence intuitively yields no additional
information about it beyond whatever Bob knew before.

To define zero knowledge mathematically we follow the following
intuition:

A proof system is zero knowledge if the verifier did not learn
anything after the interaction that he could not have learned on his
own.

Here is how we formally define this:

A proof system \((P,V)\) for \(f\) is zero knowledge if for every
efficient verifier strategy \(V^*\) there exists an efficient
probabilistic algorithm \(S^*\) (known as the simulator) such that for
every \(x\) s.t. \(f(x)=1\), the following random variables are
computationally indistinguishable:

The output of \(V^*\) after interacting with \(P\) on input \(x\).

The output of \(S^*\) on input \(x\).

That is, we can show the verifier does not gain anything from the
interaction, because no matter what algorithm \(V^*\) he uses, whatever he
learned as a result of interacting with the prover, he could have just
as equally learned by simply running the standalone algorithm \(S^*\) on
the same input.

The natural way to define security is to say that a system is secure if
some “laundry list” of bad outcomes X,Y,Z can’t happen. The definition
of zero knowledge is different. Rather than giving a list of the events
that are not allowed to occur, it gives a maximalist simulation
condition.

At its heart the definition of zero knowledge says the following:
clearly, we cannot prevent the verifier from running an efficient
algorithm \(S^*\) on the public input, but we want to ensure that this is
the most he can learn from the interaction. This simulation paradigm
has become the standard way to define security of a great many
cryptographic applications. That is, we bound what an adversary Eve can
learn by postulating some hypothetical adversary Lilith that is under
much harsher conditions (e.g., does not get to interact with the prover)
and ensuring that Eve cannot learn anything that Lilith couldn’t have
learned either. This has an advantage of being the most conservative
definition possible, and also phrasing security in positive terms-
there exists a simulation - as opposed to the typical negative terms -
events X,Y,Z can’t happen. Since it’s often easier for us to think of
positive terms, paradoxically sometimes this stronger security condition
is easier to prove. Zero knowledge is in some sense the simplest setting
of the simulation paradigm and we’ll see it time and again in dealing
with more advanced notions.

The definition of zero knowledge is confusing since intuitively one
thing that if the verifier gained confidence that the statement is true
than surely he must have learned something. This is another one of
those cases where cryptography is counterintuitive. To understand it
better, it is worthwhile to see the formal proof that the protocol above
for quadratic residousity is zero knowledge:

Protocol ZK-QR above is a zero knowledge protocol.

Let \(V^*\) be an arbitrary efficient strategy for Bob. Since Bob only
sends a single bit, we can think of this strategy as composed of two
functions:

\(V_1(x,m,x')\) outputs the bit \(b\) that Bob chooses on input \(x,m\)
and after Alice’s first message is \(x'\).

Both \(V_1\) and \(V_2\) are efficiently computable. We now need to come up
with an efficient simulator \(S^*\) that is a standalone algorithm that on
input \(x,m\) will output a distribution indistinguishable from the output
\(V^*\). The simulator \(S^*\) will work as follows:

The correctness of the simulator follows from the following claims (all
of which assume that \(x\) is actually a quadratic residue, since
otherwise we don’t need to make any guarantees and in any case Alice’s
behaviour is not well defined):

Claim 1: The distribution of \(x'\) computed by \(S^*\) is identical to
the distribution of \(x'\) chosen by Alice.

Claim 2: With probability at least \(1/2\), \(b'=b\).

Claim 3: Conditioned on \(b=b'\) and the value \(x'\) computed in step
2, the value \(s''\) computed by \(S^*\) is identical to the value that
Alice sends when her first message is \(X'\) and Bob’s response is \(b\).

Together these three claims imply that in expectation \(S^*\) only invokes
\(V_1\) and \(V_2\) a constant number of times (since every time it goes
back to step 1 with probability at most \(1/2\)). They also imply that the
output of \(S^*\) is in fact identical to the output of \(V^*\) in a true
interaction with Alice. Thus, we only need to prove the claims, which is
actually quite easy:

Proof of Claim 1: In both cases, \(x'\) is a random quadratic residue.
QED

Proof of Claim 2: This is a corollary of Claim 1; since the
distribution of \(x'\) is identical to the distribution chosen by Alice,
in particular \(x'\) gives out no information about the choice of \(b'\).
QED

Proof of Claim 3: This follows from a direct calculation. The value
\(s''\) sent by Alice is a square root of \(x'\) if \(b=0\) and of \(x'x^{-1}\)
if \(x=1\). But this is identical to what happens for \(S^*\) if \(b=b'\). QED

Together these complete the proof of the theorem.

Reference:zkqrthm is interesting but not yet good enough to guarantee
security in practice. After all, the protocol that we really need to
show is zero knowledge is the one where we repeat this procedure \(n\)
times. This is a general theorem that if a protocol is zero knowledge
then repeating it polynomially many times one after the other (so called
“sequential repetition”) preserves zero knowledge. You can think of this
as cryptography’s version of the equality “\(0+0=0\)”, but as usual,
intuitive things are not always correct and so this theorem does require
(a not super trivial) proof. It is a good exercise to try to prove it on
your own. There are known ways to achieve zero knowledge with negligible
soundness error and a constant number of communication rounds, see
Goldreich’s book (Vol 1, Sec 4.9).

Zero knowledge proof for Hamiltonicity.

We now show a proof for another language. Suppose that Alice and Bob
know an \(n\)-vertex graph \(G\) and Alice knows a Hamiltonian cycle \(C\)
in this graph (i.e.. a length \(n\) simple cycle- one that traverses all
vertices exactly once). Here is how Alice can prove that such a cycle
exists without revealing any information about it:

Protocol ZK-Ham:

Common input: graph \(H\) (in the form of an \(n\times n\) adjacency
matrix); Alice’s private input: a Hamiltonian cycle
\(C=(C_1,\ldots,C_n)\) which are distinct vertices such that
\((C_\ell,C_{\ell+1})\) is an edge in \(H\) for all
\(\ell\in\{1,\ldots,n-1\}\) and \((C_n,C_1)\) is an edge as well.

Bob chooses a random string \(z\in \{0,1\}^{3n}\)

Alice chooses a random permutation \(\pi\) on \(\{1,\ldots, n\}\) and
let \(M\) be the \(\pi\)-permuted adjacency matrix of \(H\) (i.e.,
\(M_{\pi(i),\pi(j)}=1\) iff \((i,j)\) is an edge in \(H\)). For every
\(i,j\), Alice chooses a random string \(x_{i,j} \in \{0,1\}^n\) and let
\(y_{i,j}=G(x_{i,j})\oplus M_{i,j}z\), where
\(G:\{0,1\}^n\rightarrow\{0,1\}^{3n}\) is a pseudorandom generator.
She sends \(\{ y_{i,j} \}_{i,j \in [n]}\) to Bob.

Bob chooses a bit \(b\in\{0,1\}\).

If \(b=0\) then Alice sends out \(\pi\) and the strings \(\{ x_{i,j} \}\)
for all \(i,j\); If \(b=1\) then Alice sends out the \(n\) strings
\(x_{\pi(C_1),\pi(C_2)}\),\(\ldots\),\(x_{\pi(C_n),\pi(C_1)}\) together
with their indices.

If \(b=0\) then Bob computes \(M\) to be the \(\pi\)-permuted adjacency
matrix of \(H\) and verifies that all the \(y_{i,j}\)’s were computed
from the \(x_{i,j}\)’s appropriately. If \(b=1\) then verify that the
indices of the strings \(\{ x_{i,j } \}\) sent by Alice form a cycle
and that indeed \(y_{i,j}=G(x_{i,j})\oplus z\) for every string
\(x_{i,j}\) that was sent by Alice.

Protocol ZK-Ham is a zero knowledge proof system for the language of
Hamiltonian graphs.Goldreich, Micali and Wigderson were the first to come up with a
zero knowledge proof for an NP complete problem, though the
Hamiltoncity protocol here is from a later work by Blum. We use
Naor’s commitment scheme.

We need to prove completeness, soundness, and zero
knowledge.

Completeness can be easily verified, and so we leave this to the
reader.

For soundness, we recall that (as we’ve seen before) with extremely
high probability the sets \(S_0=\{ G(x) : x\in\{0,1\}^n \}\) and
\(S_1 = \{ G(x)\oplus z : x\in\{0,1\}^n \}\) will be disjoint (this
probability is over the choice of \(z\) that is done by the verifier).
Now, assuming this is the case, given the messages \(\{ y_{i,j} \}\) sent
by the prover in the first step, define an \(n\times n\) matrix \(M'\) with
entries in \(\{0,1,?\}\) as follows: \(M'_{i,j}=0\) if \(y_{i,j}\in S_0\) ,
\(M'_{i,j}=1\) if \(y_{i,j}\in S_1\) and \(M'_{i,j}=?\) otherwise.

We split into two cases. The first case is that there exists some
permutation \(\pi\) such that (i) \(M'\) is a \(\pi\)-permuted version of
the input graph \(G\) and (ii) \(M'\) contains a Hamiltonian cycle.
Clearly in this case \(G\) contains a Hamiltonian cycle as well, and hence
we don’t need to consider it when analyzing soundness. In the other case
we claim that with probability at least \(1/2\) the verifier will reject
the proof. Indeed, if (i) is violated then the proof will be
rejected if Bob chooses \(b=0\) and if (ii) is violated then the proof
will be rejected if Bob chooses \(b=1\).

We now turn to showing zero knowledge. For this we need to build a
simulator \(S^*\) for an arbitrary efficient strategy \(V^*\) of Bob.
Recall that \(S^*\) gets as input the graph \(H\) (but not the Hamiltonian
cycle \(C\)) and needs to produce an output that is indistinguishable from
the output of \(V^*\). It will do so as follows:

Pick \(b'\in\{0,1\}\).

Let \(z\in \{0,1\}^{3n}\) be the first message computed by \(V^*\) on
input \(H\).

If \(b'=0\) then \(S^*\) computes the second message as Alice does:
chooses a random permutation \(\pi\) on \(\{1,\ldots, n\}\) and let \(M\)
be the \(\pi\)-permuted adjacency matrix of \(H\) (i.e.,
\(M_{\pi(i),\pi(j)}=1\) iff \((i,j)\) is an edge in \(H\)). In contrast,
if \(b'=1\) then \(S^*\) lets \(M\) be the all \(1'\) matrix. For every
\(i,j\), \(S^*\) chooses a random string \(x_{i,j} \in \{0,1\}^n\) and let
\(y_{i,j}=G(x_{i,j})\oplus M_{i,j}z\), where
\(G:\{0,1\}^n\rightarrow\{0,1\}^{3n}\) is a pseudorandom generator.

Let \(b\) be the output of \(V^*\) when given the input \(H\) and the
first message \(\{ y_{i,j} \}\) computed as above. If \(b\neq b'\) then
go back to step 0.

We compute the fourth message of the protocol similarly to how Alice
does it: if \(b=0\) then it consists of \(\pi\) and the strings
\(\{ x_{i,j} \}\) for all \(i,j\); If \(b=1\) then we pick a random
length-\(n\) cycle \(C'\) and the message consists of the \(n\) strings
\(x_{C'_1,C'_2}\),\(\ldots\),\(x_{C'_n,C'_1}\) together with their
indices.

Output whatever \(V^*\) outputs when given the prior message.

We prove the output of the simulator is indistinguishable from the
output of \(V^*\) in an actual interaction by the following claims:

Claim 1: The message \(\{ y_{i,j} \}\) computed by \(S^*\) is
computationally indistinguishable from the first message computed by
Alice.

Claim 2: The probability that \(b=b'\) is at least \(1/3\).

Claim 3: The fourth message computed by \(S^*\) is computationally
indistinguishable from the fourth message computed by Alice.

We will simply sketch here the proofs (again see Goldreich’s book for
full proofs):

For Claim 1, note that if \(b'=0\) then the message is identical to the
way Alice computes it. If \(b'=1\) then the difference is that \(S^*\)
computes some strings \(y_{i,j}\) of the form \(G(x_{i,j})+z\) where Alice
would compute the corresponding strings as \(G(x_{i,j})\) this is
indistinguishable because \(G\) is a pseudorandom generator (and the
distribution \(U_{3n}\oplus z\) is the same as \(U_{3n}\)).

Claim 2 is a corollary of Claim 1. If \(V^*\) managed to pick a message
\(b\) such that \(\Pr[ b=b' ] < 1/2 - negl(n)\) then in particular it could
distinguish between the first message of Alice (that is computed
independently of \(b'\) and hence contains no information about it) from
the first message of \(V^*\).

For Claim 3, note that again if \(b=0\) then the message is computed in a
way identical to what Alice does. If \(b=1\) then this message is also
computed in a way identical to Alice, since it does not matter if
instead of picking \(C'\) at random, we picked a random permutation \(\pi\)
and let \(C'\) be the image of the Hamiltonian cycle under this
permutation.

This completes the proof of the theorem.

Why is this interesting?

The reason that a protocol for Hamiltonicity is more interesting than a
protocol for Quadratic residuosity is that Hamiltonicity is an
NP-complete question. This means that for every other NP language \(L\),
we can use the reduction from \(L\) to Hamiltonicity combined with
protocol ZK-Ham to give a zero knowledge proof system for \(L\). In
particular this means that we can have zero knowledge proofs for the
following languages:

The language of numbers \(m\) such that there exists a prime \(p\)
dividing \(m\) whose remainder modulo \(10\) is \(7\).

The language of tuples \(X,e,c_1,\ldots,c_n\) such that \(c_i\) is an
encryption of a number \(x_i\) with \(\sum x_i = X\). (This is
essentially what we needed in the voting example above).

For every efficient function \(F\), the language of pairs \(x,y\) such
that there exists some input \(r\) satisfying \(y=F(x\|r)\). (This is
what we often need in the “protocol compiling” applications to show
that a particular output was produced by the correct program \(F\) on
public input \(x\) and private input \(r\).)

⊕Using a zero knowledge protocol for Hamiltonicity we can obtain a zero
knowledge protocol for any language \(L\) in NP. For example, if the
public input is a SAT formula \(\varphi\) and the Prover’s secret input is
a satisfying assignment \(x\) for \(\varphi\) then the verifier can run the
reduction on \(\varphi\) to obtain a graph \(H\) and the prover can run the
same reduction to obtain from \(x\) a Hamiltonian cycle \(C\) in \(H\). They
can then run the ZK-Ham protocol to prove that indeed \(H\) is Hamiltonian
(and hence the original formula was satisfiable) without revealing any
information the verifier could not have obtain on his
own.

Parallel repetition and turning zero knowledge proofs to signatures.

While we talked about amplifying zero knowledge proofs by running them
\(n\) times one after the other, one could also imagine running the \(n\)
copies in parallel. It is not trivial that we get the same benefit of
reducing the error to \(2^{-n}\) but it turns out that we do in the cases
we are interested in here. Unfortunately, zero knowledge is not
necessarily preserved. It’s an important open problem whether zero
knowledge is preserved for the ZK-Ham protocol mentioned above.However, Fiat and Shamir showed that in protocols (such as the ones we
showed here) where the verifier only sends random bits, then if we
replaced this verifier by a random function, then both soundness and
zero knowledge are preserved. This suggests a non-interactive version
of these protocols in the random oracle model, and this is indeed widely
used. Schnorr designed signatures based on this non interactive version.